Prove that if x is a rational number and y is an irrational number,then x+y is an irrational number. If in addition, x≠0, then show that xy is an irrational number.
As is very often the case, we do not need to write this as a proof by contradiction. We can prove the contrapositive directly.
We can prove directly:
x is rational (x+y is rational y is rational)
(using -- that is, Q is closed under subtraction)
Therefore (by contraposition of the imbedded conditional)
x is rational (y is not rational x+y is not rational)
This is logically equivalent to
(x is rational & y is not rational) x+y is not rational)
Suppose p and and r .
Prove a contradiction and conclude that if p and , then
Suppose p and . Prove that q.
Conclude that If p and , then r.
The two methods are very closely related and I don't know of anyone who accepts one and not the other. (Although many/most/all intuitionists refuse to accept either contradiction or contrapositive.)
Proof by contrapositive
Suppose that x is rational and is rational.
Then the difference is rational.
Hence is we know that y is irrational, then must have been irrational.
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